Pathwise uniform convergence of a full discretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise

TL;DR

This paper establishes a pathwise uniform convergence rate for a full discretization scheme combining Euler and finite element methods applied to a three-dimensional stochastic Allen-Cahn equation with multiplicative noise, validated by numerical experiments.

Contribution

It introduces a novel technique to prove pathwise uniform convergence of finite element discretizations for nonlinear stochastic parabolic equations in general spatial L^q-norms.

Findings

Convergence rate derived using maximal L^p-regularity estimates.

Numerical experiments confirm theoretical convergence rates.

Method applicable to nonlinear stochastic parabolic equations.

Abstract

This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization combines the Euler scheme for temporal approximation and the finite element method for spatial approximation. A pathwise uniform convergence rate is derived, encompassing general spatial \( L^q \)-norms, by using discrete versions of deterministic and stochastic maximal \( L^p \)-regularity estimates. Additionally, the theoretical convergence rate is validated through numerical experiments. The primary contribution of this work is the introduction of a technique to establish the pathwise uniform convergence of finite element-based full discretizations for nonlinear stochastic parabolic equations within the framework of general spatial \( L^q \)-norms.